FINITE DIFFERENCE ANALYSIS OF THE TRANSIENT TEMPERATURE

PROFILE WITHIN THE GHARR-1 FUEL ELEMENT

T. A. Annafi, A. A. Gyeabour I, E. H. K. Akaho, M. Annor-Nyarko and C. R. Quaye

Computational Science and Engineering, Department of Nuclear Engineering, School of Nuclear and Allied Sciences, University of Ghana, P.O Box AE 1, Accra, Ghana.

ABSTRACT

Mathematical model of the transient heat distribution within the fuel element with shutdown

heat generation rates has been developed. The shutdown heat considered were residual fission

heating and fission product decay heat. A finite difference scheme for the discretization by

implicit method was employed. A method of solution was provided and MATLAB program

was developed to outline temperature distribution within the GHARR-1 fuel element after

shutdown due to reactivity insertion accident.

The simulations showed a steady state temperature of about 341.3 K which deviated from

that reported in the GHARR-1 Safety Analysis report by 2 % error margin. The average

temperature obtained under transient condition was found to be approximately 444 K which

was lower than the melting point of 913 K for the Aluminium cladding.

Thus, the GHARR-1 fuel element was stable and there would be no release of radioactivity in

the coolant during accident conditions.

Key Words

Transient heat conduction, shutdown heat generation, residual fission power, Fission

product decay power, finite difference method, fuel element, Cladding

1. INTRODUCTION

During the steady state operation of the reactor, temperature distribution of fuel element was

determined by thermal balance between heat generated and heat conducted to the coolant [1].

Provided there are adequate heat-removal systems, the temperature in the core does not

exceed specific safety limits, and damage to the fuel pins and other reactor materials would

be prevented [2].

An accurate description of the temperature distribution in the fuel element is essential for the

prediction of the behavior of the reactor component. Also, the impact of the fuel and cladding

temperature on the neutron reaction rates provides an incentive for accurate modelling of the

temperature behavior under transient as well as steady state operating conditions [3].

The numerical method is better than the analytical solution for the estimation of the fuel

temperatures because the analytical method often treats the problem by simplifications of the

reality (in cases of very complex geometry, boundary conditions and temperature dependent

thermal properties) [4].

Transient temperature profile without heat generation rates within the fuel pin of GHARR-1

has been solved analytically using Bessel Functions [5], while the shutdown heat generation

rates (residual fission power and fission product decay power) after reactor shutdown due to

reactivity insertion accident has been estimated [6].

For the transient heat conduction problem considered, the residual fission power and fission

product decay power were the source terms. Also, the steady state temperature served as the

initial temperatures before shutdown.

The study presented in this paper was conducted to estimate the temperature variations

coupled with heat generation rates in GHARR-1 fuel rod using finite difference method.

2. THERMAL ANALYSIS FOR A CYLINDRICAL GHARR-1 FUEL ELEMENT

GHARR-1 is a tank-in-pool type, low power research MNSR, which is under-moderated, and

with natural convection cooling. The reactor employs highly enriched uranium as fuel, light

water as moderator and coolant, and metallic beryllium as reflector. The rated power of the

non-uniform thermal neutron reactor is about 30 kW and the maximum thermal neutron flux

is 1.0 x 1012 n cm-2s-1 with 10 irradiation sites (5 inside and 5 outside the beryllium annulus

reflector). The main components of GHARR-1 are the reactor vessel, reactor core (fuel

elements), beryllium reflectors, irradiation sites, control rod, and various detectors [7].

The adopted methodology in analysing transients in GHARR-1 MNSR involved solving the

heat conduction equation with heat generation at steady state by finite difference technique. A

mathematical model was then developed for transient. The shutdown heat generation rates

considered were fission product decay heat and residual fission heating

2.1 Mathematical Modelling

There is no gap between the fuel pellet and the surface of the cladding in GHARR-1 reactor

(MNSR) and the heat generated by nuclear fission is conducted through fuel meat to fuel

cladding and to the coolant. A radial heat conduction model was analyzed to determine the

transient temperature distribution within the fuel element immediately after shut down of the

reactor. The heat distribution was modelled based on the following assumptions that:

a) axial heat conduction was negligible.

b) internal heat generation considered were residual fission heat and fission product

decay heat was uniformly distributed radially over the fuel pellet cross-section.

c) all fission product decay heat were deposited locally in the fuel meat.

d) fuel element length to diameter ratio was large, hence axial temperature gradient was

small, and axial distribution of temperature remains constant.

e) one dimensional cylindrical model was adopted for calculating the temperature

distribution in the core.

f) thermal conductivity of the fuel pellet and the cladding depends on the temperature.

g) specific heat capacity, density of fuel pellet and coolant are temperature dependent.

h) the fuel meat was assumed to be homogeneous.

Fig.1 Cross section of cylindrical fuel element with coolant region.

The general form of the spatial distribution of temperature in a fuel element with internal heat

generation under transient condition is governed by the equation [8]

∇2T +qG

' ' '

k= 1

α∂T∂ t

(1)

where α is the thermal diffusivity of the material in m2/s, α= kρF c pF

, ρF is the density of the

fuel in g/m3, k is the thermal conductivity in W/mK, c pF is the specific heat capacity of the

fuel in J/m3K and qG'' ' is the rate of heat generation after shutdown in W.

But qG'' '=Pr+Pd

where

Pr is the residual fission power and Pd is the fission product decay power estimated as

Pr=P0×16141

×e−10141

t [6] (2)

where Po is operating power in W and t is the time after shutdown in s

and

Pd=P0 C t− y [6] (3)

where Po is operating power, t is the time after shutdown, C and y are constants with values

provides in [6]

For a cylinder, equation (1) becomes

1α

∂ T∂ t

=∂2T∂ r2 + 1

r∂ T∂r

+qG

' ' '

k (4)

Changing the equation into a more suitable form for finite difference solution by involving a

new variable η such that ∂ η=∂ rr

The convective heat transfer from the clad surface (at r=rf ) is governed by

−k c∂T∂ r

¿r=r f=h(T r f −T b) (5)

where T bis the mean bulk temperature in K, T rf is the clad outer surface temperature in K, k c

is the thermal conductivity in W/mK of the clad and h is the convective heat transfer

coefficient of the coolant in W/m2K.

The boundary conditions employed in solving the above equations are

1.∂ T∂r

¿r=0=0 (6)

2. k c∂T∂ r

¿r=rc=k f∂ T∂r

¿r=rc (7)

2.2 Numerical Analysis

Then ∂ T∂r

=1r

∂T∂ η

(8)

and ∂2T∂r 2 = 1

r2

∂2T∂ η2 − 1

r2

∂ T∂ η

(9)

By putting equations (8) and (9) into (4), the heat conduction within the fuel becomes

∂ T∂ t

= αr2

∂2T∂ η2 +

qG'' '

ρc p

(10)

For the clad region, the heat conduction takes place according to

ρc c pc∂T∂ t

=kc∂2T∂ r2 +kc

1r

∂T∂r

, r∈[rc , r f ] (11)

where ρc, c pc∧kc represent the density in g/m3, specific heat capacity in J/m3K and thermal

conductivity in W/mK respectively for the clad region.

rc , rf represent the inner and outer clad radius respectively as shown in the Fig. 1.

The system was treated as a set of grid points and the temperature represented as nodal

points. The dimension of each cell in the radial direction is ∆ r , and the volume is A ∆ r where

A is the cross-sectional area of the cell [9].

The temperature at various nodes and time steps for the fuel region is

(1+2α ∆ t

(r ∆ η )2 )Ti , j+1

=α ∆ t

(r ∆ η )2(T i−1 , j+1+T i+1 , j+1) +T i , j+

qG, j+1' ' '

ρcp

∆ , r → r i (12)

but r ∆ η=∆ r and F0=α ∆ t

(∆ r )2

Thus,

(1+2 F0)T i , j+1=F0 (T i−1 , j+1+T i+1 , j+1)+qG, j+1

' ' '

ρc p

∆ t+T i , j (13)

where F0 is a dimensionless number known as the Fourier number or modulus of the fuel.

Equation (13) can be written in the form

−c i T i−1+ai T i−bi T i+1=d i (14)

where

a i=(1+2 F0 ), b i=c i=F0, d i=T i , j+qG, j+1

' ' '

ρc p

∆ t

The implicit finite difference form of equation (11) which represents the temperature at each

node and time step within the clad also takes the form

T i , j+1=F0 c (T i−1 , j+1−2T i , j+1+T i+1 , J+1)+T i , j (16)

This implies

(1+2 F0c) T i , j+1=F0 c (T i−1 , j+1+T i+1 , j+1 )+T i , j (17)

where F0c is the Fourier modulus of the clad

Equation (17) can be written as

−c i T i−1+ai T i−bi T i+1=d i (18)

where

a i=(1+2 F0c ), b i=c i=F0 c, d i=T i , j

Fig. 2 One dimensional nodal points in the fuel, cladding and water regions

Fig. 2 represents the 1D grid overlay of the GHARR-1 fuel element. The fuel region was

divided in to ten equal nodes having width Δrf, while the clad region was divided into three

nodes with width Δrc, in the radial direction.

At the fuel centreline (for i = 1), equation (4) revealed that the second term in the RHS would

become infinite which is unacceptable (i.e. the second term can be written as ∂ T∂ rr

)

From the first boundary condition, at r=0, ∂ T∂r

=0

Therefore,

∂ T∂ rr

=00

(19)

Applying L’Hopital’s rule to the centre condition, we obtain

∂ T∂ rr

=

∂∂ r

( ∂ T∂ r

)

∂∂ r

(r )=

∂2 T∂ r2

1=∂2T

∂ r2 (20)

Putting equation (20) into equation (4), we get

2∂2T∂ r2 =1

α∂ T∂t

−qG

'' '

k (21)

In finite difference form, equation (21) becomes

(1+4 F0)T i , j+1=2 F0 ( T i−1 , j+1+T i+1 , j+1 )+qG , j+1

' ' '

ρcp

∆ t +T i , j (22)

Using mirror-image technique at the centre of the fuel (r= 0), equation (22) becomes

(1+4 F0)T 1 , j+1=4 F0T 2 , j+1+qG , j+1

' ' '

ρc p

∆ t +T1 , j (23)

Since T 0 , j+1=T 2 , j+1

Equation (23) was re-written as

a1T 1 , j+1=b1 T 2, j+1+d1 (24)

Similarly, putting i=2¿ N into equations (13) and (17) and applying the specified boundary

conditions to obtain an expression in the form

a iT i , j+1=b iT i+1 , j+1+c i T i−1 , j+1+d i (25)

At the convective boundary, i.e. the clad-coolant interface (for i=13¿

−k c

T13 , j+1−T 12 , j+1

∆ rc

=h(T 13−T b) (26)

where T b is the bulk coolant temperature in K

but T b=T I+T O

2 i.e. T O=2 T b−T I

where T I and T O are inlet and out outlet coolant temperatures respectively.

Therefore,

T 13=( hh+2 ρw ∆ r wcw )T 13+(

2 ρw ∆ rw cw

h+2 ρw ∆ rw cw

)T I

(27)

Putting equation (27) into equation (26), we get

[ kc

∆ rc

−h+ h2

h+2 ρw ∆ rw cw]T 13, j+1−

k c

∆ rc

T 12, j+1

¿k c

∆ rc

T 12, j+2 hρw ∆ rw cw

h+2 ρw ∆ rw cw[9] (28)

The set of simultaneous difference equations (equations (24), (25) and (28)) were expressed

in matrix notation as follows

¿ (29)

where the blank spaces in the matrix represent zeros.

The matrix above can be written as

A T new=D (30)

where T New= Ti,j+1 are the unknown transient temperatures in K to be determined and D is a

column vector of known values that contains the heat source term in J.

For each time step, T New was found by solving the tridiagonal system since all the coefficients

a i, b i, c i and d i are known.

The algorithm for the numerical computation of transient temperature distribution with heat

generation rates was developed and a computer program implemented and simulated in

Matlab version R2013a with Symbolic Math Toolbox. The Matlab code was implemented on

a 64 bit computer, with quad core processor, calibrated using published data [2] and the

output verified and validated by manual calculations.

The simulation data of the solution analysis conformed to that reported in the final safety

analysis report [7]; the temperatures generated by simulations were below limit for safe

operations of the reactor.

3. RESULTS AND DISCUSSION

Figure 4 represents the steady state temperature profile with a low power density of 380 W/m

for the GHARR-1 MNSR. The initial inlet and outlet coolant temperatures assumed where

303 K and 332 K respectively. The steady state temperature was found to be about 342 K and

the outer clad surface temperature of 313 K. The temperature served as the initial radial

temperatures for transient scenarios considered.

The temperature drop from the centre of the fuel to the surface of the clad was found to be

approximately 0.001 K. The small difference in temperature within the fuel element is

because there is no GAS gap between the fuel rod and the cladding of GHARR-1 MNSR.

0 0.5 1 1.5 2 2.5

x 10-3

315

320

325

330

335

340

345

Radial Distance in meters

Temp

eratur

e in d

egree

Kelv

in

Fig.4 Steady State Temperature Distribution within GHARR-1 fuel element

Fig. 5 represents the radial distribution up to 1500 s after reactor shutdown. The plots indicate

a parabolic thermal decay pattern from the center of the fuel to the clad-coolant surface. The

temperatures recorded both at the centerline and clad surface increased with time due to the

large value of the shutdown heat sources incorporated. The temperature range at the fuel

centre line (r = 0) was between 342 K and 448 K. At the clad-coolant interface, the

temperature recorded was between 341 K to 444 K. The increase in temperature was as a

result of heat being removed from the core at a slower rate compared to the rate at which heat

was produced.

Furthermore, the residual fission and fission product decay heat rate generated after shutdown

accounted for the temperature rise.

0 0.5 1 1.5 2 2.5

x 10-3

340

360

380

400

420

440

460

Radial Distance in meters

Tem

pera

ture

in d

egre

e Ke

lvin

T at 1 sec

T at 250sec

T at 500secT at 1000sec

T at 1500sec

Fig.5 Radial Temperature Distribution for up to 1500 s after shutdown.

0 0.5 1 1.5 2 2.5

x 10-3

442

444

446

448

450

452

Radial Distance in meters

Temp

eratur

e in d

egree

Kelv

in

T at 1000sec

T at 1250sec

T at 1500secT at 1750sec

T at 2000sec

Fig.6 Radial Transient Temperature Profile for times between 1000 s and 2000 s

Fig. 6 shows radial temperature distribution between 1000 s and 2000 s. At t = 1500 s, the

temperature rose to a maximum of about 451 K at the centre and 447 K at the clad surface

after which it decreased to approximately 450 K and 448 K at the centreline for simulation

times t = 1750 and t = 2000 s respectively.

Fig. 7 shows the transient distribution with respect to shutdown time. An initial temperature

increase from 1 to about 800 s at various nodal locations considered. The temperatures

increased with increase in reactor shutdown time because of the residual heat and decay heat

considered which increased the rate of heat production. Node 1 represents, the fuel centerline

temperature; node 5 is the distribution within the fuel; node 10 is the fuel-clad interface

temperature; node 12 shows the temperature distribution within the clad region and node 13

represents the clad-coolant interface temperature profile. Between t = 800 and 1600 s, there is

a constant temperature (asymptotic curve) before a gradual decline at t = 1750 s which

continued till the end of simulation. At t = 2000 s, a temperature of about 430 K was

recorded.

However, when a mean coolant temperature of 329 K was considered, negative reactivity

feedback balanced the core excess reactivity making the reactor subcritical.

0 200 400 600 800 1000 1200 1400 1600 1800 2000340

360

380

400

420

440

460

shutdown time in seconds

Tem

pera

ture

in d

egre

e Ke

lvin

Node 1

Node 5

Node 10Node 12

Node 13

Fig.7 Plots of transient temperature variations against shutdown time

Comparing the graph of the steady state temperature profile to that of the transient

temperature distribution as shown in Fig. 8, both curves exhibit a thermal decay pattern. The

results for the transient scenario one second after shutdown were higher than steady state

temperature values because a higher rate of heat generation was considered for the transient

scenario.

The different heat generation rates, heat transfer coefficient, clad surface and mean coolant

temperatures considered for the steady state and transient accounted for the temperature rise.

At steady state, a uniform power density of 380 W/m, heat transfer coefficient of 513W/m 2 K

and an inlet coolant temperature of 303 K were incorporated. During transient, a non-

uniform shutdown volumetric heat generation rate was considered for every temperature

profile computed at each time step.

0 0.5 1 1.5 2 2.5

x 10-3

340.5

341

341.5

342

342.5

343

343.5

Radial Distance in meters

Tem

pera

ture

in d

egre

e Ke

lvin

Transient

Steady State

Fig.8 A graph of steady state and transient temperature profile against distance.

The sources of error encountered were as a result of truncation errors, computational errors

and round off errors due to finite word length. These errors cause the deviation of the

simulated results from the actual results reported as reported in the SAR [7]. However, the

program was run with half the number of step size, Δr (or larger N), and time steps, Δt which

indicates the solution for t =2000 s is sufficiently accurate.

Statistical analysis was done by computing the mean, standard deviation, maximum (peak)

and minimum (clad surface) temperature values from the generated steady state temperature

distribution in Fig. 4

Table 1 Statistical Analysis of the Steady State temperature Profile in GHARR-1 Fuel Element

CALCULATED EXPERIMENTED

Maximum Mean Minimum RMS SD GHARR-1 PARR-2

Temperatur

e (K )

341.7 341.3 340.8 0.1204 0.36 343.0 345

RE1 0.02 0.05

RE1 is Relative Error = Absolute Error/Actual Value

where PARR-2 is Pakistan Research Reactor 2, RMS is the root mean square and SD is the

standard deviation.

Steady state temperature values deviated from a mean value of 341.3 K by a standard

deviation of 0.36 (approximately one standard deviation). The peak fuel temperature of 341.7

K was 2% lower than the reported value of 343 K for GHARR-1[2] and 5% lower than that

reported in PARR-2 [9] similar to GHARR-1 MNSR.

5. CONCLUSIONS

Transient heat conduction models for the GHARR-1 fuel element have been implemented.

Discretization of the fuel pellet and cladding has been defined for thirteen nodes of equal

sizes. No gas gap was considered between the fuel rod and clad. Numerical flow chat and

solution algorithms were developed for the transient heat conduction equation and Matlab

code developed to simulate the model.

The temperature profile after reactor shutdown showed parabolic decay pattern for every time

step simulated. Temperature increased at both the centreline and at the clad surface initially

because the rate of heat production was greater than heat removal from the system due to

transient. The temperatures decreased after some time (t = 1500 s) due to cooling natural by

natural convection. Also, the heat generation rates considered decreased with time making the

temperature recorded at clad surface decrease.

According to the final safety analysis of GHARR-1, the reactor was operated at relatively low

temperatures of 288 K – 333 K and in the event of an accident the cladding temperature is

expected to be 363 K. However, the computed results of peak centreline and cladding

temperatures ranged between 332.003 K – 332.002 K and 411.600 K - 404.200 K

respectively. The recorded fuel temperature of approximately 411.6 K is below the melting

point of the U-Al alloy which was about 913 K. Local meltdown of the fuel meat cannot

occur and fission gases cannot pass through the meat.

Thus, for the transient scenario considered, the GHARR-1 MNSR can be adjudged to be

operating in safe mode to forestall the incident of release of radioactive in to the coolant.

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